Under recent circulars on the notions of convexity for real sets and functions like $E$-convexity and $(E,F)$-convexity, we expand the notions of $(E,F)$ and semi- $(E,F)$-convexity to include domains and functions in complex space. We examine their properties and interrelationships. As a consequence, we apply the associated results on a non-linear semi- $(E,F)$-convex programming problem with cone-constraints in complex space. We discuss the existence and uniqueness of its optimal solution and establish the necessary and sufficient conditions for a feasible point to be an optimal solution to such a problem. The related results in real space can be deduced as special cases.